Question

Derive an expression for the mass ( M_p ) of the planet in terms of ( a ), ( d ), and ( g ), the universal gravitational constant. Assume that the gravitational effect of the star at the planet's surface is negligible.

1. a. ( M_p = a^2 ⋅ gG ⋅ d )
2. b. ( M_p = G ⋅ a^2g ⋅ d )
3. c. ( M_p = a ⋅ gG ⋅ d^2 )
4. d. ( M_p = G ⋅ ag ⋅ d^2 )

Answer 1

To find the mass of a planet, you utilize Newton's law of universal gravitation and the formula g = G*M_p/d^2, then solve for M_p to get M_p = g*d^2/G.

Step-by-step explanation:

To derive an expression for the mass (M_p) of the planet in terms of the acceleration due to gravity at the planet's surface (g), the universal gravitational constant (G), and the distance between the center of the planet and the object (d), which is essentially the planet's radius in this context, we utilize Newton's law of universal gravitation. This law expresses the force (F) of gravity as F = G * (m * M_p) / d^2, where (m) is the mass of the object experiencing the gravitational force. Since we are interested in the acceleration at the planet's surface, we can substitute F with m * g, which gives us m * g = G * (m * M_p) / d^2. After canceling out (m) on both sides, the equation simplifies to g = G * M_p / d^2. Finally, solving for (M_p) gets us the mass of the planet, M_p = g * d^2 / G.